Hadronic uncertainties versus new physics for the W boson mass and Muon g − 2 anomalies

There are now two single measurements of precision observables that have major anomalies in the Standard Model: the recent CDF measurement of the W mass shows a 7σ deviation and the Muon g − 2 experiment at FNAL confirmed a long-standing anomaly, implying a 4.2σ deviation. Doubts regarding new physics interpretations of these anomalies could stem from uncertainties in the common hadronic contributions. We demonstrate that these two anomalies pull the hadronic contributions in opposite directions by performing electroweak fits in which the hadronic contribution was allowed to float. The fits show that including the g − 2 measurement worsens the tension with the CDF measurement and conversely that adjustments that alleviate the CDF tension worsen the g − 2 tension beyond 5σ. This means that if we adopt the CDF W mass measurement, the case for new physics in either the W mass or muon g − 2 is inescapable regardless of the size of the SM hadronic contributions. Lastly, we demonstrate that a mixed scalar leptoquark extension of the Standard Model could explain both anomalies simultaneously.

The CDF collaboration at Fermilab recently reported the world's most precise direct measurement of the W boson mass, M CDF W = 80:4335 ± 0:0094 GeV 1 , based on 8.8/fb of data collected between 2002-2011. This deviates from the Standard Model (SM) prediction by about 7σ. The recent FNAL E989 measurement of the muon's anomalous magnetic moment furthermore implies a new world average of a μ = 16 592 061(41) × 10 −11 2 , which is in 4.2σ tension with the SM theory prediction from the Muon g − 2 Theory Initiative, 116 591 810(43) × 10 −11 3 . This prediction is based on results from refs.  .
Whilst the Fermilab g − 2 measurement was in agreement with the previous BNL E821 measurement 30 , as shown in Fig. 1 there appears to be tension between the new CDF measurement and previous measurements, including the previous CDF measurement with only 2.2/fb of data 31 . Updates to systematic uncertainties shift the previous measurement by 13.5 MeV, however, such that the CDF measurements are self-consistent. In the Supplementary Note 1 we find a reduced chisquared from a combination of N = 7 measurements of about χ 2 / (N − 1) ≃ 3 and a tension of about 2.5σ. Nevertheless, we show that these two measurements could point towards physics beyond the SM with a common origin and, under reasonable assumptions, that the new CDF W mass measurement pulls common hadronic contributions in a direction that significantly strengthens the case for new physics in muon g − 2.
The predictions also suffer from parametric uncertainties, with the largest uncertainties coming from m t and may be around 9 MeV 64 , and depend on estimates of the hadronic contributions to the running of the fine structure constant, Δα had Δα ð5Þ had ðM 2 Z Þ, defined at the scale M Z for five quark flavours. This is constrained by electroweak (EW) data and by measurements of the e + e − → hadrons cross section (σ had ) through the principal value of the integral 65 where m π 0 is the neutral pion mass. Turning to muon g − 2, the SM prediction for a μ includes hadronic vacuum polarization (HVP) and hadronic light-by-light (HLbL) contributions in addition to the QED and EW contributions that can be calculated perturbatively from first principles 3 . Although HVP is not the main contribution for a μ , it suffers from the largest uncertainty and it is hard to pin down its size. The HLbL contributions in contrast have a significantly smaller uncertainty, with data-driven methods now providing the most precise estimates but with lattice QCD results that are consistent with these and which also contribute to the final result in ref. 3 . Two approaches are commonly used to extract the contributions from HVP. First, a traditional data-driven method in which the HVP contributions are determined from measurements of σ had using the relationship 68 where m μ and m π 0 are the muon and neutral pion masses, respectively, and K(s) is the kernel function as shown in refs. 68,69 . This approach results in a HVP μ = 693:1ð4:0Þ × 10 À10 with an uncertainty of <0.6% [8][9][10]12,13,70 . The second approach uses lattice QCD calculations. The recent leading-order lattice QCD calculations for HVP from the BMW collaboration significantly reduced the uncertainties and resulted in a HVP μ = 707:7ð5:5Þ × 10 À10 71 . This, however, shows tension with the σ had measurements method.
The M W and muon g − 2 calculations are in fact connected by the fact that both Δα had and the HVP contributions can be extracted from the hadronic cross section, σ had ð ffiffi ffi s p Þ, through eqs. (2) and (3). We assume that the energy dependence of this cross-section, gð ffiffi ffi s p Þ, is reliably known for ffiffi ffi s p ≥ m π 0 9,12 , but that the overall scale, σ had , may be  adjusted, This simple modification is similar to scenario (3) in ref. 65 . There are of course more complicated possibilities, including increases and decreases in the hadronic cross section at different energies. Reference 72 considered these complicated possibilities to be implausible, though this is a somewhat subjective matter; see Supplementary Note 2 for further discussion. Using eqs. (2) and (4) we may trade σ had for Δα had giving Δα had ∝ σ had . The HVP contributions depend on Δα had and conversely estimates of the HVP contributions from either hadronic cross-sections or lattice QCD constrain Δα had . Further details of the transformation between Δα had and a HVP μ are provided in the Supplementary Note 2. Thus we can transfer constraints on Δα had from measurements of M W to constraints on the HVP contributions to muon g − 2 and vice-versa 72-74 through global EW fits.
In this work, we study how the new M W measurement from CDF impacts estimates of muon g − 2 in global EW fits and show that a common explanation of muon g − 2 and the CDF M W from hadronic uncertainties are not possible. Then we demonstrate that in contrast a scalar leptoquark model could provide a simultaneous explanation of both muon g − 2 and the W mass anomalies.
We further scrutinize the impact of assumptions about the HVP contributions and the W mass through several fits shown in Table 1. In the first three fits, the W mass is only indirectly constrained by EW data, and Δα had is constrained by the BMWc determination of the HVP contributions, by the e + e − data, and indirectly by EW data. The second and final three fits are similar, though the W mass is constrained by the PDG 2021 world average and by the CDF 2022 measurement, respectively. In each case we show the overall goodness of fit, and how much the bestfit muon g − 2 and W mass predictions deviate from the world average and the recent CDF measurement, respectively. Regardless of the constraints imposed on Δα had , including the CDF measurement results in poor overall goodness of fit and increases the tension between the SM prediction for g − 2 and the world average. The tension between the SM prediction for the W mass and the CDF measurement range from 3.2σ to 7.8σ. However, the former occurs only when estimates of HVP are completely ignored (final column) and at the expense of increased tension in the SM g − 2 prediction and a poor overall goodness of fit. Note that this includes the scenario where we do not include any input values for M W or Δα had in the global EW fit, as shown in the third data column (of twelve). Even in this case there is still a large tension with the CDF measurement (5.8σ), indicating that other EW observables also constrain Δα had . Using the e + e − estimates of HVP, which is a standard choice, we see about 5σ tension in both g − 2 and the W mass. In fact, the CDF measurement takes the tension between the SM prediction for muon g − 2 and the measurements slightly beyond 5σ. Switching to BMWc estimates of HVP partially alleviates the tension in g − 2 but results in increased tension with the CDF W mass measurement.
In summary, our fits showed the extent to which the new W mass measurement worsens tension with muon g − 2, using the reasonable assumption that the energy-dependence of the hadronic cross section that connects these is well-known and not modified by for example very light new physics. The anomalies pull Δα had in opposite directions in EW fits, making it even harder to explain both within the SM. We thus now turn to a new physics explanation.

Interpretation in Scalar Leptoquark model
Even without light new physics, sizable BSM contributions to muon g − 2 can be obtained by an operator that gives an internal chirality flip in the one-loop muon g − 2 corrections (see e.g. refs. 80,81 for a review). On the other hand, BSM contributions to the W mass can be obtained when there are large corrections to the oblique parameter T 82 . We show that a scalar leptoquark model can satisfy both of these criteria and provide a simultaneous explanation of both muon g − 2 and the W mass anomalies. We anticipate other possibilities, including composite models with non-standard Higgs bosons 83 .
Scalar leptoquarks (LQs) (see ref. 84 for a review), or more specifically the scalar leptoquarks referred to as S 1 ð3,1,1=3Þ and R 2 (3, 2, 7/6) in refs. [85][86][87] , are well known to provide the chirality flip needed to give a large contribution to a μ 88 , and have also been proposed for a simultaneous explanation of the flavour anomalies 89 . Furthermore due to the mass splitting between its physical states the SU(2) doublet R 2 is capable of making a considerable contribution to the W mass. However we find that the mass splitting from a conventional Higgs portal interaction cannot generate corrections big enough to reach the CDF measurement, unless the interaction λ R 2 H R y 2 HH y R 2 is non-perturbative. We thus analyze the plausibility of situations in which one-loop contributions to the anomalous muon magnetic moment and W mass corrections are created via the mixing of two scalar LQs through the Higgs portal. For simplicity, we consider the S 1 &S 3 ð3,3,1=3Þ scenario, where the first term is responsible for the mixing of the two LQs, and the second specifies the interaction between quarks and leptons L mix = λH y τ ! ÁS 3 HS * 1 + h:c: ð6Þ Although a coupling between S 1 and the left-handed lepton and quark fields is also allowed, we do not initially consider it here. Instead we show that it is possible to have new physics explanations of the CDF 2022 measurement and the 2021 combined a μ world average that originate from the same feature of the model, namely the combination of the S 1 and S 3 states through a non-vanishing mixing parameter, λ.
For simplicity, we assume that only the couplings to muons that give the large chirality flipping enhancement from muon g − 2 i.e., y tμ R and y bμ L are non-vanishing in the new Yukawa coupling. After EW symmetry breaking, we have four scalar LQs, one with an electromagnetic charge Q = 4/3, one with Q = − 2/3 and two with Q = 1/ 3. The Q = 1/3 states mix through the λ interaction resulting in mass eigenstates S ± 1=3 + and S ± 1=3 À with masses m S + and m S À : where ϕ is the mixing angle. The masses m S 3 , m S 1 and the mixing parameter λ can be obtained from m S + , m S À and ϕ from δ = λv 2 2 = s ϕ c ϕ ðm 2 S À À m 2 S + Þ ð9Þ where v = 246 GeV is the vacuum expectation value. We also define Δm m S + À m S À as the mass splitting between the two mass eigenstates S + and S − . This mass splitting generates a non-vanishing oblique correction to the T parameter at one-loop 90 , This function vanishes when the masses are degenerate, that is, lim m 1 !m 2 Fðm 1 ,m 2 Þ = 0. When Δm = 0, the custodial symmetry is restored, and the corrections to the T parameter vanish as m S 3 = m + = m À . The shift in M W from the SM prediction can be related to the oblique T parameter via, where c W and s W are the cosine and sine of the Weinberg angle. There are, furthermore, contributions from S and U that are subdominant in our LQ model. We determine the T that is required to explain the CDF 2022 measurement from our EW global fits and use that in combination with eq. (12) to test if LQ scenarios can explain this data. We checked analytically and numerically that our calculation obeys decoupling, with the additional BSM contributions approaching zero in the limit of large LQ masses. We cross-checked eq. (12) with a full one-loop calculation of the T parameter using SARAH 4.14.3 91 , FeynArts 3.11 92 , FormCalc 9.9 93 and LoopTools 2.16 94 , finding good agreement with the results using just eq. (12). With the same setup we also verified that the combined contributions from S and U to M W are small and do not impact significantly on our results. Finally we also implemented this model in FlexibleSUSY 95-98 using the same SARAH model file and the recently updated M W calculation 98 and again found reasonable agreement with the results of our analysis described above. Whilst the mass splitting impacts the W mass, the mixing impacts muon g − 2. Indeed, the mixing between interaction eigenstates allows the physical mass eigenstates to have both left-and right-handed couplings to muons and induces chirality flipping enhancements to muon g − 2 90 and we simplify our notation by letting y L y bμ L and y R y tμ R . Note that in this case there is a cancellation between the contribution of the lighter and heavier mass eigenstates, which reduces the effect of the very large chirality flipping enhancement m t /m μ somewhat. If we consider couplings between S 1 state and left-handed muons as well, the contributions would be considerably enhanced, so this would simply make it easier to explain a μ while having little or no impact on the W mass prediction. BSM contributions to a μ and the W mass both require nonvanishing Δm. For a μ , it further requires non-vanishing mixing ϕ and relies on y L y R . Thus it is possible to find explanations of both a μ and the 2022 CDF measurement of M W by varying y L y R and Δm with non-zero mixing angle ϕ. In Fig. 3 we show regions in the Δm-ffiffiffiffiffiffiffiffiffi ffi y L y R p plane that explain both measurements, where we have fixed the LQ mass to 1.7 TeV, a little above the LHC limit, and we also fixed the mixing angle ϕ = − π/8. LQ couplings of greater than about a half can explain the a μ measurement within 1σ when we use the SM prediction from the theory white paper, where e + e − data is used for a HVP μ . Explaining the SM prediction from the BMW collaboration requires even smaller couplings, though in this case the tension with the SM is anyway <2σ. Using e + e − data to also fix Δα had means there then remains an additional deviation between the SM M W prediction and the measured values. To explain the new 2022 CDF result with BSM contributions as well, Δm ≈ 75 GeV is then required, and the dual explanation of the M W and a μ anomalies can be achieved in the region where the green CDF 2022 band overlaps with the light blue e + e − band in Fig. 3. The deviation between the SM prediction for the W boson mass and the 2021 PDG value is not so large and within 2σ it does not need new physics contributions, so the yellow 2σ band for this in Fig. 3 can extend to Δm ≈ 0, but to within 1σ a small non-zero Δm is required. Further, the interaction coupling λ is proportional to the mass splitting Δm with fixed mixing angle ϕ. In order to keep the coupling perturbative (|λ| < ffiffiffiffiffiffi 4π p ), there is an upper limit on the mass splitting as shown by the grey band in Fig. 3. Note that the region that can accommodate the CDF measurement is close to the non-perturbative region, as the CDF measurement requires large mass splitting. However, it is still possible to explain the new M W measurement within 1σ in the perturbative region.

Further constraints
This model establishes a proof of principle of a simple, dual explanation of both anomalies. There remains, however, the question of whether this model or extensions of it can simultaneously explain recent flavour physics measurements and anomalies and satisfy additional phenomenological constraints. The latter may be particularly severe as the required Yukawa couplings are Oð1Þ.
For example, the recently measured branching ratio BR(h → μμ) 99,100 can be an important probe of leptoquark explanations of muon g − 2 101,102 . Reference 101 showed that when you have S 1 and S 3 with only right-handed couplings for S 1 there is already a significant tension with the current measurements. There are several ways to avoid this tension. If the leptoquarks are embedded in a more fundamental theory there could be additional light states that result in cancellations with the leptoquark contribution to h → μμ, for example through destructive interference between tree-and loop-level diagrams. This can be achieved by extending the LQ model in the framework of the two-Higgs-doublet model in the wrong-sign Yukawa coupling region 103 . Alternatively we can reintroduce the left-handed coupling of the S 1 state, which brings two benefits.
First, allowing significant left-handed couplings from S 1 substantially reduces the size of the Yukawa couplings needed to explain muon g − 2 (as stated earlier). We show in the Supplementary Note 3 that, as can be anticipated from ref. 101 , this makes it possible to satisfy the BR(h → μμ) data while simultaneously explaining muon g − 2, while keeping the mass splitting fixed to a value required to explain the CDF measurement of the W mass. Second introducing this coupling gives additional freedom that can help explain the well-known anomalies of Lepton Flavour Universality Violation, while avoiding other limits from flavour physics.
Indeed, the severe constraints from μ → eγ, a e , etc. can all be evaded by allowing the proper flavour ansatz for the Yukawa couplings 104 . At the same time the R K * and R D * anomalies can be explained via extra tree-level processes from the scalar LQ with Q = 4/3 plane of our LQ model that predict the W-boson mass and muon g − 2 in agreement with measurements. The mixing angle ϕ is set to be − π/8 and m S À = 1:7 TeV. The grey region is excluded by the perturbativity requirement |λ| < ffiffiffiffiffiffi 4π p .
Article https://doi.org/10.1038/s41467-023-36366-7 to b → sμ + μ − and two scalar LQs with Q = 1/3 to b → cτν. Although the latter two scalar LQs also contribute to R νν K * through tree-level process b → sνν, these enhancements are under control and their effects are not in conflict with the current measurement 104 .
Finally, we show that an explanation of the W mass in our model must be accompanied by new physics below about 2 TeV. In order to explain the CDF measurement at 2σ with S = 0 (which is a good approximation in our model), we need 0.14 ≲ T ≲ 0.17. Expanding eq. (12), such that combining the lower limit on the T parameter and the perturbativity limit λ ≤ ffiffiffiffiffiffi 4π p we obtain As the mass splitting, Δm = m + − m − ≲ 100 GeV, our model thus predicts two Q = 1/3 LQ states below about 2 TeV.

Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on request.

Code availability
The custom computer codes used to generate results are available from the corresponding author on request.